PAPER
www.rsc.org/softmatter | Soft Matter
A theory of constrained swelling of a pH-sensitive hydrogel†‡
Romain Marcombe,ab Shengqiang Cai,a Wei Hong,c Xuanhe Zhao,a Yuri Lapustab and Zhigang Suo*a
Received 20th August 2009, Accepted 23rd November 2009
First published as an Advance Article on the web 26th January 2010
DOI: 10.1039/b917211d
Many engineering devices and natural phenomena involve gels that swell under the constraint of hard
materials. The constraint causes a field of stress in a gel, and often makes the swelling inhomogeneous even
when the gel reaches a state of equilibrium. This paper develops a theory of constrained swelling of
a pH-sensitive hydrogel, a network of polymers bearing acidic groups, in equilibrium with an aqueous
solution and mechanical forces. The condition of equilibrium is expressed as a variational statement of the
inhomogeneous field. A free-energy function accounts for the stretching of the network, mixing of the
network with the solution, and dissociation of the acidic groups. Within a Legendre transformation, the
condition of equilibrium for the pH-sensitive hydrogel is equivalent to that for a hyperelastic solid. The
theory is first used to compare several cases of homogenous swelling: a free gel, a gel attached to a rigid
substrate, and a gel confined in three directions. To analyze inhomogeneous swelling, we implement
a finite element method in the commercial software ABAQUS, and illustrate the method with a layer of
the gel coated on a spherical rigid particle, and a pH-sensitive valve in microfluidics.
1. Introduction
Immersed in an aqueous solution, a network of covalently crosslinked polymers imbibes the solution and swells, resulting in
a hydrogel. The amount of swelling is affected by mechanical forces,
pH, salt, temperature, light, and electric field.1,2 Gels are being
developed for diverse applications as actuators, converting nonmechanical stimulations to large displacements and appreciable
forces.3–6 Many applications require that the gels swell against the
constraint of hard materials. For example, a microfluidic valve
involves a gel anchored by a rigid pillar, and the gel swells in
response to a change in the pH, blocking the flow.7 Analogous
mechanisms have been used by plants to regulate microscopic flow,8
and in oilfields to enhance production.9 As another example, an
array of rigid rods embedded in a gel rotate when the humidity in the
environment drops below a critical value.10,11 It has also been
appreciated that, in a spinal disc, the swelling of the nucleus
pulposus is constrained by the annulus fibrosus, and that
understanding this constrained swelling is central to developing
a synthetic hydrogel to replace damaged nucleus pulposus.12
Despite the ubiquity of constrained swelling in practice, the
theory of constrained swelling requires substantial work to be
broadly useful in analyzing engineering devices and natural
phenomena. Developers of methods of analysis face two essential
challenges. First, swelling of a gel is affected by a large number of
a
School of Engineering and Applied Sciences, Harvard University,
Cambridge, MA 02138, USA. E-mail: suo@seas.harvard.edu
b
IFMA-LAMI, French Institute of Advanced Mechanics, Campus de
Clermont-Ferrand/Les C
ezeaux, 63175 Aubi
ere, France
c
Department of Aerospace Engineering, Iowa State University, Ames, IA
50011, USA
† This paper is part of a Soft Matter themed issue on Emerging Themes in
Soft Matter: Responsive and Active Soft Materials. Guest Editors: Anna
C. Balazs and Julia Yeomans.
‡ Electronic supplementary information (ESI) available: Further
equations and the code of user subroutine for pH-sensitive hydrogels.
See DOI: 10.1039/b917211d
784 | Soft Matter, 2010, 6, 784–793
stimuli. It is unrealistic to expect any single material model to
describe the behavior of many gels. Second, when a gel is constrained by a hard material, the swelling often induces in the gel an
inhomogeneous field of stress and large deformation. The
magnitude of the stress is of central importance to applications
such as valves and actuators. The large deformation, in addition to
being important to applications, may also lead to cavities, creases,
buckles, and other intriguing patterns that are hard to analyze.13–17
Following a recent trend in the study of inhomogeneous
deformation of complex materials, we have been pursuing
a modular approach, which in effect meets the two challenges
separately. As an example, we have shown that the swelling of
a neutral network in equilibrium is equivalent to the deformation
of a hyperelastic material.18 The latter can be readily analyzed by
adding a material model to commercial finite element software
like ABAQUS. This approach is applicable to a neutral network
characterized by a free-energy function of any form. Commercial
software like ABAQUS is widely used in many fields of engineering, and has been developed to analyze large deformation of
extraordinary complexity. Consequently, this approach has
enabled researchers to use the commercial software to analyze
complex phenomena in gels.19,20
The present paper goes beyond the neutral network, and
develops a theory for a pH-sensitive hydrogel, a network of
polymers bearing acidic groups, in equilibrium with an aqueous
solution and a set of mechanical forces. Following our recent
work on polyelectrolyte gels,21 we express the condition of
equilibrium as a variational statement. For a pH-sensitive gel,
the variational statement includes the following fields: the
displacement of the network, the concentrations of the solvent
and ions, and the degree of acidic dissociation. The variations are
subject to auxiliary conditions of several types, including the
conservation of various species, incompressibility of molecules,
and electroneutrality in the gel and in the external solution.
Our task in the present paper is greatly simplified by the
assumption of electroneutrality. To appreciate this assumption,
This journal is ª The Royal Society of Chemistry 2010
consider a highly charged network immersed in a dilute solution
of ions, so that the concentration of the counterions in the gel
exceeds that in the external solution. At the interface between the
gel and the external solution, the counterions in the gel spill into
the external solution, and the region near the interface is no
longer neutral, leading to an electric double layer of a thickness
scaled by the Debye length. Outside the electric double layer,
electroneutrality is nearly maintained in the gel and in the
external solution. In many applications, the Debye length is
much smaller than other lengths of interest. This paper will not
be concerned with the electric double layer, and will assume that
the gel is electroneutral. This assumption will miss phenomena at
the size scale comparable to the Debye length, but will capture
the overall behavior of the gel.21
As a model material, the gel is characterized by a free-energy
function developed by Flory,22 Recke and Tanaka,23 BrannonPeppas and Peppas,24 and others. (Incidentally, these authors
also assumed electroneutrality.) The free-energy function
accounts for the stretching of the network, mixing of the network
and the solution, and dissociation of the acidic groups. The
model is used to compare several cases of homogeneous swelling:
a free gel, a gel attached to a rigid substrate, and a gel confined in
three directions.
Inhomogeneous swelling is then studied by developing a finite
element method. Inhomogeneous swelling of pH-sensitive gels
has been studied in several recent papers,25–27 but the existing
methods have not been demonstrated for the analysis of complex
phenomena of large deformation. In this paper, we represent the
free energy as a functional of the field of deformation by using
a Legendre transformation. Within this representation, the
inhomogeneous field in a pH-sensitive hydrogel in equilibrium is
again equivalent to the field in a hyperelastic solid. We implement the finite element method by writing a user-supplied
subroutine in the commercial software ABAQUS, and illustrate
the method with a layer of the gel coated on a spherical rigid
particle, and a pH-sensitive valve in microfluidics. We hope that
this work will enable other researchers to study complex
phenomena in pH-sensitive hydrogels. To this end, we have made
our code freely accessible online.28
2. The condition of equilibrium for inhomogeneous
swelling
Fig. 1 sketches a model system: a network of covalently crosslinked polymers bearing acidic groups AH. When the network
imbibes the solvent, some of the acidic groups dissociate into
hydrogen ions H+ mobile in the solvent, and conjugate bases A
attached to the network. Once dissociated, the conjugate base A
gives rise to a network-attached charge, i.e., a fixed charge. The
reaction is reversible:
AH 4 A + H+
This journal is ª The Royal Society of Chemistry 2010
The external solution is composed of four species: solvent
molecules (i.e., water), hydrogen ions, counterions that bear
charges of the sign opposite to the fixed charges (e.g., sodium
ions), and co-ions that bear charges of the same sign as the fixed
charges (e.g., chloride ions). To describe essentials of the method
of analysis, we neglect the dissociation of water, and assume that
counterions and co-ions are monovalent. Let nS, nH , n+ and n
be the numbers of particles of the four species in the external
solution. When these numbers change by small amounts, the free
energy of the external solution changes by
+
mSd
nS + mH d
nH + m+d
n+ + md
n
+
+
(2.3)
where mS, mH , m+ and m are the electrochemical potentials of the
four species in the external solution. The external solution is in
a state of equilibrium, so that the electrochemical potential of
each species is homogeneous in the external solution.
Fig. 2 illustrates a gel undergoing inhomogeneous swelling. We
take the stress-free dry network as the state of reference. A small
part of the network is named after the coordinate of the part, X,
when the network is in the state of reference. Let dV(X) be an
element of volume, dA(X) be an element of area, and NK(X) be
the unit vector normal to the element of area.
+
(2.1)
The three species equilibrate when their concentrations satisfy
þ H
A
(2.2)
¼ Ka
½AH
where Ka is the constant of acidic dissociation.
Fig. 1 A network of polymers imbibes a solution and swells, resulting in
a gel. The polymers are covalently crosslinked and bear acidic groups, some
of which dissociate into hydrogen ions mobile in the solvent, and fixed
charges attached to the network. The external solution is composed of four
mobile species: solvent molecules, hydrogen ions, counterions, and co-ions.
Fig. 2 A dry network is taken to be the state of reference. In the current
state, the network is immersed in an aqueous solution and subject to a set
of mechanical forces.
Soft Matter, 2010, 6, 784–793 | 785
In the current state, the part of the network X moves to a place
with coordinate x. The function
xi ¼ xi(X)
(2.4)
describes a field of deformation. The deformation gradient of the
network is
vxi ðXÞ
FiK ¼
(2.5)
vXK
In the current state, let Bi(X)dV(X) be the external mechanical
force applied on the element of volume, and Ti(X)dA(X) be the
external mechanical force applied on the element of area. When
the network deforms by a small amount, dxi(X), the field of
mechanical force does work
Ð
Ð
BidxidV + TidxidA
(2.6)
Following a common practice in formulating a field theory, we
stipulate that an inhomogeneously swollen gel can be divided
into many small volumes, and each small volume is locally in
a state of homogeneous swelling, characterized by a nominal
density of free energy W as a function of various thermodynamic
variables. Consequently, the Helmholtz free energy of the gel in
the current state is given by
Ð
WdV
(2.7)
The gel, the external solution, and the mechanical forces
together constitute a thermodynamic system, held at a fixed
temperature. The Helmholtz free energy of the system is the sum
of the free energy of the gel, the free energy of the external
solution, and the potential energy of the mechanical forces.
When the system is in equilibrium, associated with small variations of the fields, the variation of the Helmholtz free energy
vanishes. Consequently, the condition of equilibrium is
Ð
dWdV +Ð mSd
nS + mHÐ d
nH + m+d
n+ + md
n
BidxidV TidxidA ¼ 0
+
+
(2.8)
Note that W is a function of various thermodynamic variables,
so that the variational statement (2.8) includes variations of
the following inhomogeneous fields: the displacement of the
network, the concentrations of the solvent and ions, and the
degree of acidic dissociation. The variations are subject to
auxiliary conditions of several types, including the conservation
of various species, incompressibility of molecules, and electroneutrality in the gel and in the external solution. These auxiliary
conditions are discussed below.
Denote the nominal concentration of species a by Ca(X). That
is, Ca(X)dV(X) is the number of particles of species a in the
element of the network when the gel is in the current state. Of the
four mobile species, the solvent molecules, the counterions, and
the co-ions are each conserved. The gel gains these particles at the
expense of the external solution:
Ð
Ð
Ð
dCS(X)dV + d
nS ¼ 0
(2.9)
n+ ¼ 0
dC+(X)dV + d
(2.10)
n ¼ 0
dC(X)dV + d
(2.11)
786 | Soft Matter, 2010, 6, 784–793
The mobile hydrogen ions, however, are not conserved, but are
produced as the acidic groups dissociate. The change in the total
number of the hydrogen ions in the system equals the change in
the number of the fixed charges:
Ð
Ð
dCH (X)dV + d
nH ¼ dCA (X)dV
+
+
(2.12)
The sum of the number of the associated acidic groups AH and
that of the fixed charges A equal the total number of the acidic
groups:
CAH(X) + CA (X) ¼ f/v
(2.13)
where f is the number of acidic groups attached to the network
divided by the total number of monomers in the network, and v is
the volume per monomer.
As discussed in Introduction, we assume that electroneutrality
prevails both in the gel and in the external solution, so that
CH (X) + C+(X) ¼ CA (X) + C(X)
(2.14)
nH + n+ ¼ n
(2.15)
+
+
Because typically the stress in a gel is small and the amount of
swelling is large, we assume that individual polymers and solvent
molecules are incompressible. Furthermore, the concentrations
of ions are assumed to be low, so that their contributions to the
volume of the gel are negligible. Under these simplifications,
when the dry network of unit volume imbibes CS number of
solvent molecules and swells to a gel of volume detF, these
volumes satisfy the condition
1 + vSCS ¼ detF
(2.16)
where vS is the volume per solvent molecule. This molecular
incompressibility is assumed in all theoretical papers cited above.
Subject to the auxiliary conditions (2.9)–(2.16), the state of the
inhomogeneously swollen gel is specified by the following independent fields: xi(X), C+(X), C(X), and CH (X). We stipulate
that the nominal density of free energy is a function:
+
W ¼ W(F,C+,C,CH ).
+
(2.17)
Using the auxiliary conditions (2.9)–(2.16), we rewrite the
condition of equilibrium (2.8) in terms of variations of the
independent fields, namely,
ð
v
vW mS
HiK detF þ Bi dxi dV
vXK vFiK vS
ð vW mS
HiK detF NK Ti dxi dA
þ
vFiK vS
ð
vW
ðmþ mH þ Þ dCþ dV
þ
vCþ
ð
vW
ðm þ mHþ Þ dC dV
þ
vC
ð
vW
dC þ dV ¼ 0
þ
vCHþ H
(2.18)
In writing (2.18), we have used the divergence theorem, as well
as an identity vdetF/vFiK ¼ HiKdetF, where HiK is the transpose
This journal is ª The Royal Society of Chemistry 2010
of the inverse of the deformation gradient, namely, HiKFiL ¼ dKL
and HiKFjK ¼ dij.
Inspecting (2.18), we write
siK ¼
vW ðF; Cþ ; C ; CHþ Þ mS
HiK det F
vFiK
vS
(2.19)
m þ mHþ ¼ kT log
The quantity siK is known as the tensor of nominal stress. The
term containing ms is due to the assumed molecular incompressibility.
The statement (2.18) holds for arbitrary variations of the
independent fields, xi(X), C + (X), C(X), and CH (X). Consequently, each line of (2.18) leads to the condition of a partial
equilibrium with respect to the variation of a single independent
field. The first line of (2.18) leads to
+
vsiK
þ Bi ¼ 0
vXK
mþ mH ¼ kT log
þ
(2.20)
(2.21)
for elements on the surface of the gel. These two equations
constitute the familiar conditions of mechanical equilibrium with
respect to the variation dxi.
The next two lines of (2.18) lead to
vW ðF; Cþ ; C ; CHþ Þ
¼ mþ mHþ
vCþ
(2.22)
!
(3.1)
Hþ
cref
þ c
c cHþ
!
ref
cref
cHþ
+
mS ¼ kTvS(
cH + c+ + c).
+
(2.23)
These equations are the conditions of ionic equilibrium with
respect to the variations in the concentrations of the counterions
and co-ions in the gel. The combinations m+ mH and m + mH
are due to the assumed electroneutrality.
The last line of (2.18) leads to
+
vW ðF; Cþ ; C ; CHþ Þ
¼0
vCHþ
+
(2.24)
This equations is the condition of chemical equilibrium with
respect to the dissociation of the acidic groups, a condition that
reproduces (2.2), as shown in the next section.
3. A specific material model
The conditions of equilibrium described in the previous section
are independent of models of the external solution and gel. This
section applies the conditions of equilibrium to a commonly used
material model.
External solution
Let c+, c and cH be the true concentration of the three species of
ions in the external solution. We assume that the external solution is dilute, so that the electrochemical potentials of the ions
relate to the concentrations as21
+
This journal is ª The Royal Society of Chemistry 2010
(3.3)
Eqn (3.1)–(3.3) express the electrochemical potential in terms
of the concentrations of the four mobile species.
pH-sensitive gel
Following Flory,22 Ricke and Tanaka,23 Brannon-Peppas and
Peppas,24 and many others, we adopt an idealized model,
assuming that the free-energy density of the gel is a sum of several
contributions:
W ¼ Wnet + Wsol + Wion + Wdis
vW ðF; Cþ ; C ; CHþ Þ
¼ m þ mHþ
vC
(3.2)
where kT is the temperature in the unit of energy, and cref
a is
a reference value of the concentration of a species.
Imagine that the solution is separated from a reservoir of pure
solvent by a membrane, which allows solvent molecules to pass
through, but not the ions. The solvent molecules will permeate
from the reservoir into the solution, until the solution is under
a pressure, the osmotic pressure, kT(
cH + c+ + c. Consequently,
relative to the pure solvent, the solvent molecules in the ionic
solution has the chemical potential
for elements in the interior of the gel. The second line of (2.18)
leads to
siKNK ¼ Ti
cþ cref
Hþ
(3.4)
where Wnet is due to stretching the network, Wsol mixing the
solvent with the network, Wion mixing ions with the solvent, and
Wdis dissociating the acidic groups.
The free energy of stretching the network is taken to be
Wnet ¼ ½NkT[FiKFiK 3 2log(detF)]
(3.5)
where N is the number of polymer chains divided by the volume
of the dry network.
The free energy of mixing the polymers and the solvent takes
the form:
kT
1
c
ðdet F 1Þlog 1 Wsol ¼
(3.6)
vS
det F
det F
This contribution consists of the entropy of mixing of the
polymers and the solvent molecules, as well as the enthalpy of
mixing, characterized by a dimensionless parameter c.
The concentrations of the mobile ions are taken to be low, so
that their contribution to the free energy is due to the entropy of
mixing, namely,
!
!
"
CHþ
Cþ
1 þ Cþ log ref
1
Wion ¼ kT CHþ log ref
cHþ det F
cþ det F
!#
C
1
þ C log ref
c det F
(3.7)
The contribution due to the dissociation of the acidic groups is
taken to be
Soft Matter, 2010, 6, 784–793 | 787
#
CA
CAH
þ CAH log
¼ kT CA log
CA þ CAH
CA þ CAH
"
Wdis
þ gCA
(3.8)
The expression consists of the entropy of dissociation and the
enthalpy of dissociation, where g is the increase in the enthalpy
when an acidic group dissociates. Note that CA and CAH are the
nominal concentration of the fixed charges and of associated
acidic groups, respectively. They are not among the independent
variables chosen to represent the free-energy function, (2.17).
Using (2.13) and (2.14), however, we can express them in terms of
the chosen independent variables, CA ¼ CH + C+ C, CAH ¼
f/v (CH + C+ C).
cHþ ðcHþ þ cþ c Þ
ðf =vÞðdet FÞ1 ðcHþ þ cþ c Þ
¼ NA Ka
(3.15)
This condition reproduces (1.2), with the identification
g
exp (3.16)
NA Ka ¼ cref
Hþ
kT
+
+
Parameters used in numerical calculations
In numerical calculations, we assume that the volume per
monomer equals the volume per solvent molecule, v ¼ vS. Electroneutrality in the external solution requires that c ¼ c+ + cH
Consequently, the composition of the external solution is specified by two independent numbers, say, the concentration of the
counterions c+ and the concentration of the hydrogen ions cH .
The later relates to the pH of the external solution,
cHþ ¼ NA 10pH .
The polymers are specified by several parameters. Recall that
N is the number of polymer chains per unit volume of the dry
network, so that 1/Nv is the number of monomers per polymer
chain. The dimensionless parameter c measures the enthalpy of
mixing the polymers and the solvent. The number f is the number
of acidic groups on a polymer chain divided by the total number
of monomers on the chain. For applications that prefer gels with
large swelling ratios, materials with low values of Nv and c and
high value of f are used. In numerical calculations, we set Nv ¼
103, c ¼ 0.1, and f ¼ 0.05. The constant of acidic dissociation,
Ka, has the same dimension as the concentration (in the unit mol
L1). We set pKa ¼ log10Ka ¼ 4.3, a commonly accepted value
for the dissociation of carboxylic acids.
We will normalize the chemical potential by kT, and normalize
the stresses by kT/v. A representative value of the volume per
molecule is v ¼ 1028 m3. At room temperature, kT ¼ 4 1021 J
and kT/v ¼ 4 107 Pa. The elastic modulus of the dry network is
NkT. For Nv ¼ 103, the elastic modulus is NkT ¼ 4 104 Pa.
+
+
Equilibrium between the gel, external solution, and mechanical
forces
Recall that the number of particles of species a in the gel in the
current state divided by the volume of the dry network defines the
nominal concentration of the species, Ca. The same number
divided by the volume of the gel in the current state defines the
true concentration of the species, ca. The two definitions are
related as Ca ¼ cadetF. Recall that when the number of particles
is counted in units of the Avogadro number, NA ¼ 6.023 1023,
the molar concentration of the species a is designated by [a]; for
example, cH ¼ NA[H+].
Recall a relation in continuum mechanics connecting the true
stress sij and the nominal stress: sij ¼ siKFjK/detF, so that (2.19)
can be written as
FjK vW mS
dij
(3.9)
sij ¼
det F vFiK vS
+
Using the function W(F,C+,C,CH ) specified above, (3.9)
becomes that
+
sij ¼
NkT FiK FjK dij ðPsol þ Pion Þdij
det F
(3.10)
4. Several cases of homogenous swelling
where
Pion ¼ kT(cH + c+ + c cH c+ c)
( )
kT
1
1
c
log 1 Psol ¼ þ
þ
vS
det F
det F ðdet FÞ2
+
+
(3.11)
(3.12)
Here Pion is the osmotic pressure due to the imbalance of the
number of ions in the gel and in the external solution, and Psol is
the osmotic pressure due to mixing the network and the solvent.
Condition (3.9) is readily interpreted: in equilibrium, the applied
stress sij equals the contractile stress of the network minus the
osmotic pressure.
The conditions of ionic equilibrium (2.22) and (2.23) become
c+/
c+ ¼ cH /
cH
+
+
c ¼ (cH /
cH )1
c/
+
+
(3.13)
(3.14)
These conditions are known as the Donnan equations. The
condition of chemical equilibrium with respect to acidic dissociation (2.24) becomes that
788 | Soft Matter, 2010, 6, 784–793
The material model described above is now applied to several
cases of homogeneous swelling (Fig. 3). In each case, the
conditions of equilibrium (3.10)–(3.15) form a set of simultaneous nonlinear algebraic equations. Their solutions illustrate
the basic behavior of a gel with or without constraint. These cases
of homogeneous swelling also provide tests for the finite element
program to be developed in the following section.
In the case of a free gel, Fig. 3a, all components of stress
vanish, and the swelling is isotropic: F ¼ ldiK. Fig. 4a plots the
swelling ratio of the gel, l3, as a function of the composition of
external solution. The latter is specified by pH, and the molar
concentration of the counterions, c+/NA. The gel swells more
when the external solution has low concentrations of both the
hydrogen ions and the counterions, but swells less when the
external solution is concentrated with either species. These trends
are considered in some detail below.
Fig. 4b plots the swelling ratio as a function of pH at a fixed concentration of the counterions. The trend can be understood in terms of the two limits: fully-associated limit and
fully-dissociated limit. When pH\\pKa , the abundance of
This journal is ª The Royal Society of Chemistry 2010
hydrogen ions causes all the acidic groups to be associated,
namely,
CAH ¼ f/v, CA ¼ 0.
(4.1)
Consequently, the network is neutral, and ions of every species
are equally distributed in the gel and the external solution:
cH ¼ cH , c+ ¼ c+, c ¼ c.
+
+
(4.2)
The balanced ions do not contribute to osmosis, Pion ¼ 0.
When pH..pKa , the scarcity of hydrogen ions causes all the
acidic groups to be dissociated, namely,
CAH ¼ 0, CA ¼ f/v
(4.3)
Fig. 3 Several cases of homogeneous swelling. (a) Free swelling. (b)
Swelling subject to biaxial constraint. (c) Swelling under triaxial
constraint.
Consequently, the network bears a known number of fixed
charges. These fixed charges must be neutralized by counterions,
as dictated by electroneutrality. Consequently, mobile ions will
be more concentrated in the gel than in the external solution.
These unbalanced ions contribute to osmosis, Pion > 0, so that
the network in the fully-dissociated limit will imbibe more
solvent than the network in the fully-associated limit.
Fig. 4c plots the swelling ratio as a function of the molar
concentration of the counterions in the external solution, c+ /NA,
at several values of pH. When pH ¼ 2, the hydrogen ions are
abundant, and the gel approaches the fully-associated limit.
When pH ¼ 9, the hydrogen ions are scarce, and the gel
approaches the fully-dissociated limit. These two limits have
been discussed above. The external solution with an intermediate
value, pH ¼ 5, deserves additional comments.20 The Donnan
equation, c+/
c+ ¼ cH /
cH , requires that the two species of positive
ions in the gel and in the external solution be distributed proportionally. When c+ < cH in the external solution, c+ < cH in
the gel. The abundance of hydrogen ions in the gel causes the
acidic groups to be mostly associated, so that the network is
nearly neutral. As c+ increases while cH is fixed, more counterions will be available in the gel, and more acidic group will
dissociate. This process of ion exchange causes the swelling ratio
to increase with the concentration of the counterions in the
external solution. When the external solution has a very high
concentration of the counterions, however, the gel behaves like
a neutral gel, and the swelling ratio drops.
Fig. 3b illustrates a layer of a gel attached to a rigid substrate.
The substrate is flat, and the thickness of the gel is much smaller
than the length and the width of the gel, so that the deformation
of the gel is homogeneous. The two stretches in the plane of the
layer is prescribed to be l0. When the gel is brought into contact
with the external solution, the two in-plane stretches remain
fixed, but the gel swells in the direction normal to the layer, with
stretch l. The swelling ratio of the substrate-attached gel varies
with the composition of the external solution, with the trends
similar to that of the unconstrained gel. However, the amount of
swelling of the free gel is significantly larger than that of the
substrate-attached gel (Fig. 5). Consequently, the amount of
swelling cannot be specified as a material property, but must be
solved as a part of the boundary-value problem.
Fig. 3c illustrates a layer of a gel attached to a rigid substrate,
with equal stretches prescribed in the plane, lT. The layer is also
constrained in the normal direction, but with a different level of
+
+
+
+
+
Fig. 4 Numerical results for a free swelling gel. (a) The swelling ratio is
plotted as a function of the two variables that specify the composition of
the external solution: the pH and the salt concentration (i.e., molar
concentration of the counterions). (b) The swelling ratio is plotted as
a function of pH for a fixed salt concentration. (c) The swelling ratio is
plotted as a function of the salt concentration at several values of pH.
This journal is ª The Royal Society of Chemistry 2010
Soft Matter, 2010, 6, 784–793 | 789
stretch lN. The gel develops a state of triaxial stress, sT and sN.
As mentioned in the Introduction, in many applications of the
pH-sensitive hydrogels, the gel has to exert a pressure on the
constraining hard material. In such applications, various ways to
change the blocking stress sN are important. Fig. 6 plots the
blocking stress as a function of the pH of the external solution at
several values of the lateral stretch. The blocking stress also
exhibits two limits. When the pH value in the external solution is
low, the abundant hydrogen ions cause the acidic groups on the
network approach the fully associated limit, and the magnitude
of the blocking stress is small. When the pH value in the external
solution is high, the scarce hydrogen ions cause the acidic groups
on the network approach the fully dissociated limit, and the
magnitude of the blocking stress is large. The magnitude of the
blocking stress can be changed by prescribing a different value of
the in-plane stretch. As expected, the magnitude of the blocking
stress increases when the lateral stretch decreases.
The theory outlined in this paper describes many of the
qualitative trends observed experimentally. However, a quantitative comparison between the theory and experiments is difficult
for several reasons. First, to highlight the essential ideas of the
theory, we have used relatively simple models of solutions. Most
experiments are carried out using more complex systems, such as
copolymers and solutions of multiple species. Second, the existing experiments often report insufficient details, leaving many
parameters to fit. With these difficulties in mind, we leave more
extensive comparison to future work, and restrict ourselves here
to a comparison between the theory and one set of experiments,
as follows.
Eichenbaum et al.28 have done a series of experiments to study
the effect of crosslink density on the swelling behavior of pHsensitive hydrogels. Ref. 28 provided all material parameters
except f in our model. The comparison is plotted in Fig. 7. The
theoretical predictions fit well with Eichenbaum’s experimental
data for poly(methacrylic acid-co-acrylic acid) gels with four
different crosslink density with one fitting parameter f ¼ 0.35.
Both the theoretical predictions and experimental results show
swelling ratio induced by the change of the pH value in outer
solution is reduced as the crosslink density increases.
Fig. 5 The swelling ratio of a free gel and a substrate-attached gel as
a function of the pH of the external solution.
790 | Soft Matter, 2010, 6, 784–793
Fig. 6 The blocking stress as a function of the pH of the external
solution at several values of the lateral stretch.
5. Finite element method
The condition of equilibrium of a pH-sensitive hydrogel is
expressed as the variational statement (2.8), which governs the
following independent inhomogeneous fields: xi(X), C+(X),
C(X), and CH (X). This variational statement has a form
different from that used in commonly available commercial finite
element software. In this section, we transform this variational
statement into a different form, which can be readily implemented in commercial finite element software.
Following a commonly used approach in thermodynamics, we
^ by a Legendre transintroduce another free-energy function W
formation:
+
^ ¼ W (m+ mH )C+ (m + mH )C mSCS
W
+
+
(5.1)
Fig. 7 Comparison between theoretical predictions and experimental
results. The scattered dots are experimental data and different lines are
the calculation results. Material parameters are given in ref. 28: salt
concentration is 0.03M, c ¼ 0.45 + 0.4894, Ka ¼ 104.7. The mole fraction
of pH sensitive monomers f ¼ 0.35 is used by fitting the calculation with
experimental data.
This journal is ª The Royal Society of Chemistry 2010
We can solve the nonlinear algebraic eqn (3.13)–(3.15), and
express CH , C+ and C in terms of cH , c+ and detF; see part A in
^ can be expressed as a function of the
the ESI.‡ Consequently, W
following independent variables:
+
+
^ ¼W
^ (F,
W
cH ,cI_+).
(5.2)
+
The physical significance of this change of variables is understood as follows. When a network is immersed in a solution, so
long as the amount of the gel is small compared to the amount of
the external solution, the composition of the external solution
remains unchanged as the gel swells. Consequently, concentrations of the hydrogen ions and counterions in the external
solution, cI_H and cI_+, remain fixed, and so do the electrochemical potentials of all the species. Inserting (5.1) into (2.18),
the condition of equilibrium (2.18) becomes that
+
Ð
Ð
Ð
^ dV ¼ BidxidV + TidxidA
dW
inhomogeneously. Near the outer surface, the gel is nearly
unconstrained, and the swelling ratio approaches that of a free
gel. Near the interface between the gel and the core, however, the
gel is constrained, and the swelling ratio is much below of that of
the free gel.
The constraint of the rigid particle also causes in the gel a field
of stress. Fig. 8b plots the distribution of the stress in the gel.
Near the outer surface of the gel, the radial stress vanishes
because of the boundary condition, and the magnitude of the
hoop stress is small because the gel is nearly free. Near the
interface between the gel and the rigid core, the radial stress is
tensile and the hoop stress is compressive. These trends can be
readily understood. If the rigid particle were removed, the gel
would swell homogeneously and stress-free, and both the inner
radius and outer radius would increase. In the presence of the
rigid particle, however, the inner radius is constrained to be of
the original size, leading to the tensile radial stress and
(5.3)
The variational statement (5.3) takes the same form as that of
a hyperelastic solid. That is, the work done by the mechanical
forces equals the variation in the free energy. Because the
composition of the external solution, cH and c+, remain fixed
when the mechanical forces do work, the variation in the free
^ ¼W
^ (F,
energy W
cH ,
c+) is entirely due to the variation of the
deformation gradient. Consequently, the variational statement
(5.3) can be readily implemented in commercial finite element
software.
We have implemented the above theory in the commercial
^ ¼
finite element software, ABAQUS, by coding the function W
^ (F,
c+) into a user-defined subroutine for a hyperelastic
W
cH ,
material. Details in implementing the finite element method may
be found in our paper on neutral gels,18 and part A of the ESI‡ of
the present paper. The subroutine is given in the part C of the
ESI‡ and posted online.29
We first test our finite element program against the cases of
homogeneous swelling. For example, Fig. 5 plots the swelling
ratios of a free gel and a substrate-attached gel. We have also
tested other cases of homogeneous swelling. In all cases, the
results obtained by the finite element method agree well with
those of the analytical solutions.
We then test the finite element program using a case of inhomogeneous swelling: a layer of a gel coated on a rigid spherical
particle (Fig. 8). The core–shell structure is immersed in a solution. When the pH of the external solution changes, the gel swells
or deswells, but the rigid particle remains inert. In this particular
calculation, when pH ¼ 2, the gel is taken to be stress-free, and
the ratio of the outer radius of the gel to the radius of the rigid
particle is set to be B/A ¼ 1.5. When pH ¼ 6, the gel swells
subject to the constraint of the rigid particle. Consequently,
a field of stress develops in the gel and the amount of swelling is
inhomogeneous, even when the gel reaches a state of equilibrium.
To compare with the finite element solution, Part B of the ESI‡
lists the differential equations for this spherical symmetric
boundary-value problem. These equations are solved by using
a finite difference method. The results are compared with those
obtained by using the finite element method.
Fig. 8a plots the distribution of the swelling ratio in the gel.
Due to the constraint of the rigid particle, the gel swells
+
+
+
This journal is ª The Royal Society of Chemistry 2010
Fig. 8 Swelling of a gel coated on a rigid spherical particle. (a) Distribution of the concentration of water in the gel. (b) Distribution of the
radial stress and hoop stress in the gel.
Soft Matter, 2010, 6, 784–793 | 791
compressive hoop stress. As shown in Fig. 8b, the results
obtained by using finite element method agree well with those
obtained by solving the ordinary differential equations.
As another illustration of the finite element method, consider
the microfluidic valve7 mentioned in the Introduction. Fig. 9
illustrates a gel coated on a rigid pillar in a microfluidic channel.
The gel is taken to deform under the plane strain conditions.
When pH ¼ 2, the gel is in a stress-free state, and the channel is
open. When pH ¼ 6, the gel swells to push against the walls of
the channel, and the channel is closed. In the open state, the outer
radius of the gel should be small to ease the flow. In the closed
state, the size of the contact between the gel and a wall, as well as
the pressure in the contact, should be large to block the flow. In
this case, the calculation needs to deal with the inhomogeneous
deformation of the gel, as well as the contact between the gel and
the walls. An analytical solution too this problem is unavailable.
However, by implementing our subroutine in ABAQUS, we can
use almost all the functions already embedded in this commercial
software.
Fig. 9 plots the deformed configuration of the valve, as well as
the size of the contact and the distribution of the pressure. We fix
the radius of the pillar, A/D ¼ 0.1. As the outer radius of the gel
increases, both the size of the contact and the pressure in the
contact increase. The size of the contact and the pressure may be
crucial for such a design for valves. In the original design of the
valve, several pillars were placed across the width of the channel.7
In such a design, the pillars form a periodic array, and the above
analysis remains valid. The finite element program may be used
to explore other patterns of pillars, or other designs of pHsensitive valves.
6. Concluding remarks
This paper develops a theory of a network of covalently crosslinked polymers bearing acidic groups, in equilibrium with an
aqueous solution and a set of mechanical forces. The inhomogeneous swelling is affected by the pH and salinity of the external
solution, as well as by the geometry of the constraint. The
condition of equilibrium is expressed as a variational statement
that governs the following independent fields: the displacement
of the network, and the concentrations of the hydrogen ions,
counterions and co-ions. By using the Legendre transformation,
the variational statement is cast into a form such that a swollen
gel in equilibrium is governed by the same equations as those for
an equivalent hyperelastic material. The theory is implemented
as a finite element method in the commercial software ABAQUS,
and is illustrated with cases of homogeneous and inhomogeneous
swelling. It is hoped that this work will enable other researchers
to study complex phenomena in pH-sensitive hydrogels. To this
end, we have made our code freely accessible online.29
Acknowledgements
This work was supported by the NSF through a grant on Soft
Active Materials (CMMI–0800161), by the DARPA through
a contract on Programmable Matter (W911NF-08- 1-0143), and
by Schlumberger through a contract on Swelling Elastomers for
Applications in Oilfields.
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Fig. 9 In a microfluidic channel, a gel is anchored by a rigid pillar. When
pH ¼ 2, the gel shrinks, and the channel is open. When pH ¼ 6, the gel
swells, and the channel is closed. As the outer radius of the gel increases,
both the size of the contact and the pressure in the contact increase.
792 | Soft Matter, 2010, 6, 784–793
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